Question

Giải phương trình:

(2017 - x)³ + (2019 - x)³ + (2x - 4036)³ = 0

Answer 1

Đặt \(2017-x=a;2019-x=b;2x-4036=c\)

\(\Rightarrow a+b+c=0\)

Do \(a+b+c=0\Rightarrow a+b=-c\Leftrightarrow\left(a+b\right)^3=-c^3\)

Có : \(a^3+b^3+c^3=\left(a+b\right)^3-3ab\left(a+b\right)+c^3=-c^3-3ab.\left(-c\right)+c^3=3abc\)

Do \(\left(2017-x\right)^3+\left(2019-x\right)^3+\left(2x-4036\right)^3=0\)

\(\Rightarrow3\left(2017-x\right)\left(2019-x\right)\left(2x-4036\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}2017-x=0\\2019-x=0\\2x-4036=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=2017\\x=2019\\x=2018\end{matrix}\right.\)

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